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"Keep in mind that for two symbols P and Q, P => Q is false only when P is `True` and Q is `False`.\n",
"Example usage of `tt_entails()`:"
]
},
{
"cell_type": "code",
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{
"data": {
"text/plain": [
"True"
]
},
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}
],
"source": [
"tt_entails(P & Q, Q)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"P & Q is True only when both P and Q are True. Hence, (P & Q) => Q is True"
]
},
{
"cell_type": "code",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"False"
]
},
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"tt_entails(P | Q, Q)"
]
},
{
"cell_type": "code",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"False"
]
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"output_type": "execute_result"
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],
"source": [
"tt_entails(P | Q, P)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"If we know that P | Q is true, we cannot infer the truth values of P and Q. \n",
"Hence (P | Q) => Q is False and so is (P | Q) => P."
]
},
{
"cell_type": "code",
"metadata": {},
"outputs": [
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"data": {
"text/plain": [
"True"
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"source": [
"(A, B, C, D, E, F, G) = symbols('A, B, C, D, E, F, G')\n",
"tt_entails(A & (B | C) & D & E & ~(F | G), A & D & E & ~F & ~G)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"We can see that for the KB to be true, A, D, E have to be True and F and G have to be False.\n",
"Nothing can be said about B or C."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Coming back to our problem, note that `tt_entails()` takes an `Expr` which is a conjunction of clauses as the input instead of the `KB` itself. \n",
"You can use the `ask_if_true()` method of `PropKB` which does all the required conversions. \n",
"Let's check what `wumpus_kb` tells us about $P_{1, 1}$."
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"(True, False)"
]
},
"metadata": {},
"output_type": "execute_result"
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],
"source": [
"wumpus_kb.ask_if_true(~P11), wumpus_kb.ask_if_true(P11)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Looking at Figure 7.9 we see that in all models in which the knowledge base is `True`, $P_{1, 1}$ is `False`. It makes sense that `ask_if_true()` returns `True` for $\\alpha = \\neg P_{1, 1}$ and `False` for $\\alpha = P_{1, 1}$. This begs the question, what if $\\alpha$ is `True` in only a portion of all models. Do we return `True` or `False`? This doesn't rule out the possibility of $\\alpha$ being `True` but it is not entailed by the `KB` so we return `False` in such cases. We can see this is the case for $P_{2, 2}$ and $P_{3, 1}$."
]
},
{
"cell_type": "code",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"(False, False)"
]
},
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"wumpus_kb.ask_if_true(~P22), wumpus_kb.ask_if_true(P22)"
]
},
{
"cell_type": "markdown",
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"### Proof by Resolution\n",
"Recall that our goal is to check whether $\\text{KB} \\vDash \\alpha$ i.e. is $\\text{KB} \\implies \\alpha$ true in every model. Suppose we wanted to check if $P \\implies Q$ is valid. We check the satisfiability of $\\neg (P \\implies Q)$, which can be rewritten as $P \\land \\neg Q$. If $P \\land \\neg Q$ is unsatisfiable, then $P \\implies Q$ must be true in all models. This gives us the result \"$\\text{KB} \\vDash \\alpha$ <em>if and only if</em> $\\text{KB} \\land \\neg \\alpha$ is unsatisfiable\".<br/>\n",
"This technique corresponds to <em>proof by <strong>contradiction</strong></em>, a standard mathematical proof technique. We assume $\\alpha$ to be false and show that this leads to a contradiction with known axioms in $\\text{KB}$. We obtain a contradiction by making valid inferences using inference rules. In this proof we use a single inference rule, <strong>resolution</strong> which states $(l_1 \\lor \\dots \\lor l_k) \\land (m_1 \\lor \\dots \\lor m_n) \\land (l_i \\iff \\neg m_j) \\implies l_1 \\lor \\dots \\lor l_{i - 1} \\lor l_{i + 1} \\lor \\dots \\lor l_k \\lor m_1 \\lor \\dots \\lor m_{j - 1} \\lor m_{j + 1} \\lor \\dots \\lor m_n$. Applying the resolution yeilds us a clause which we add to the KB. We keep doing this until:\n",
"\n",
"* There are no new clauses that can be added, in which case $\\text{KB} \\nvDash \\alpha$.\n",
"* Two clauses resolve to yield the <em>empty clause</em>, in which case $\\text{KB} \\vDash \\alpha$.\n",
"\n",
"The <em>empty clause</em> is equivalent to <em>False</em> because it arises only from resolving two complementary\n",
"unit clauses such as $P$ and $\\neg P$ which is a contradiction as both $P$ and $\\neg P$ can't be <em>True</em> at the same time."
]
},
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{
"cell_type": "markdown",
"metadata": {},
"source": [
"There is one catch however, the algorithm that implements proof by resolution cannot handle complex sentences. \n",
"Implications and bi-implications have to be simplified into simpler clauses. \n",
"We already know that *every sentence of a propositional logic is logically equivalent to a conjunction of clauses*.\n",
"We will use this fact to our advantage and simplify the input sentence into the **conjunctive normal form** (CNF) which is a conjunction of disjunctions of literals.\n",
"For eg:\n",
"<br>\n",
"$$(A\\lor B)\\land (\\neg B\\lor C\\lor\\neg D)\\land (D\\lor\\neg E)$$\n",
"This is equivalent to the POS (Product of sums) form in digital electronics.\n",
"<br>\n",
"Here's an outline of how the conversion is done:\n",
"1. Convert bi-implications to implications\n",
"<br>\n",
"$\\alpha\\iff\\beta$ can be written as $(\\alpha\\implies\\beta)\\land(\\beta\\implies\\alpha)$\n",
"<br>\n",
"This also applies to compound sentences\n",
"<br>\n",
"$\\alpha\\iff(\\beta\\lor\\gamma)$ can be written as $(\\alpha\\implies(\\beta\\lor\\gamma))\\land((\\beta\\lor\\gamma)\\implies\\alpha)$\n",
"<br>\n",
"2. Convert implications to their logical equivalents\n",
"<br>\n",
"$\\alpha\\implies\\beta$ can be written as $\\neg\\alpha\\lor\\beta$\n",
"<br>\n",
"3. Move negation inwards\n",
"<br>\n",
"CNF requires atomic literals. Hence, negation cannot appear on a compound statement.\n",
"De Morgan's laws will be helpful here.\n",
"<br>\n",
"$\\neg(\\alpha\\land\\beta)\\equiv(\\neg\\alpha\\lor\\neg\\beta)$\n",
"<br>\n",
"$\\neg(\\alpha\\lor\\beta)\\equiv(\\neg\\alpha\\land\\neg\\beta)$\n",
"<br>\n",
"4. Distribute disjunction over conjunction\n",
"<br>\n",
"Disjunction and conjunction are distributive over each other.\n",
"Now that we only have conjunctions, disjunctions and negations in our expression, \n",
"we will distribute disjunctions over conjunctions wherever possible as this will give us a sentence which is a conjunction of simpler clauses, \n",
"which is what we wanted in the first place.\n",
"<br>\n",
"We need a term of the form\n",
"<br>\n",
"$(\\alpha_{1}\\lor\\alpha_{2}\\lor\\alpha_{3}...)\\land(\\beta_{1}\\lor\\beta_{2}\\lor\\beta_{3}...)\\land(\\gamma_{1}\\lor\\gamma_{2}\\lor\\gamma_{3}...)\\land...$\n",
"<br>\n",
"<br>\n",
"The `to_cnf` function executes this conversion using helper subroutines."
]
},
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"<div class=\"highlight\"><pre><span></span><span class=\"k\">def</span> <span class=\"nf\">to_cnf</span><span class=\"p\">(</span><span class=\"n\">s</span><span class=\"p\">):</span>\n",
" <span class=\"sd\">"""Convert a propositional logical sentence to conjunctive normal form.</span>\n",
"<span class=\"sd\"> That is, to the form ((A | ~B | ...) & (B | C | ...) & ...) [p. 253]</span>\n",
"<span class=\"sd\"> >>> to_cnf('~(B | C)')</span>\n",
"<span class=\"sd\"> (~B & ~C)</span>\n",
"<span class=\"sd\"> """</span>\n",
" <span class=\"n\">s</span> <span class=\"o\">=</span> <span class=\"n\">expr</span><span class=\"p\">(</span><span class=\"n\">s</span><span class=\"p\">)</span>\n",
" <span class=\"k\">if</span> <span class=\"nb\">isinstance</span><span class=\"p\">(</span><span class=\"n\">s</span><span class=\"p\">,</span> <span class=\"nb\">str</span><span class=\"p\">):</span>\n",
" <span class=\"n\">s</span> <span class=\"o\">=</span> <span class=\"n\">expr</span><span class=\"p\">(</span><span class=\"n\">s</span><span class=\"p\">)</span>\n",
" <span class=\"n\">s</span> <span class=\"o\">=</span> <span class=\"n\">eliminate_implications</span><span class=\"p\">(</span><span class=\"n\">s</span><span class=\"p\">)</span> <span class=\"c1\"># Steps 1, 2 from p. 253</span>\n",
" <span class=\"n\">s</span> <span class=\"o\">=</span> <span class=\"n\">move_not_inwards</span><span class=\"p\">(</span><span class=\"n\">s</span><span class=\"p\">)</span> <span class=\"c1\"># Step 3</span>\n",
" <span class=\"k\">return</span> <span class=\"n\">distribute_and_over_or</span><span class=\"p\">(</span><span class=\"n\">s</span><span class=\"p\">)</span> <span class=\"c1\"># Step 4</span>\n",
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{
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"`to_cnf` calls three subroutines.\n",
"<br>\n",
"`eliminate_implications` converts bi-implications and implications to their logical equivalents.\n",
"<br>\n",
"`move_not_inwards` removes negations from compound statements and moves them inwards using De Morgan's laws.\n",
"<br>\n",
"`distribute_and_over_or` distributes disjunctions over conjunctions.\n",
"<br>\n",
"Run the cell below for implementation details."
]
},
{
"cell_type": "code",
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"\n",
" </style>\n",
"</head>\n",
"<body>\n",
"<h2></h2>\n",
"\n",
"<div class=\"highlight\"><pre><span></span><span class=\"k\">def</span> <span class=\"nf\">eliminate_implications</span><span class=\"p\">(</span><span class=\"n\">s</span><span class=\"p\">):</span>\n",
" <span class=\"sd\">"""Change implications into equivalent form with only &, |, and ~ as logical operators."""</span>\n",
" <span class=\"n\">s</span> <span class=\"o\">=</span> <span class=\"n\">expr</span><span class=\"p\">(</span><span class=\"n\">s</span><span class=\"p\">)</span>\n",
" <span class=\"k\">if</span> <span class=\"ow\">not</span> <span class=\"n\">s</span><span class=\"o\">.</span><span class=\"n\">args</span> <span class=\"ow\">or</span> <span class=\"n\">is_symbol</span><span class=\"p\">(</span><span class=\"n\">s</span><span class=\"o\">.</span><span class=\"n\">op</span><span class=\"p\">):</span>\n",
" <span class=\"k\">return</span> <span class=\"n\">s</span> <span class=\"c1\"># Atoms are unchanged.</span>\n",
" <span class=\"n\">args</span> <span class=\"o\">=</span> <span class=\"nb\">list</span><span class=\"p\">(</span><span class=\"nb\">map</span><span class=\"p\">(</span><span class=\"n\">eliminate_implications</span><span class=\"p\">,</span> <span class=\"n\">s</span><span class=\"o\">.</span><span class=\"n\">args</span><span class=\"p\">))</span>\n",
" <span class=\"n\">a</span><span class=\"p\">,</span> <span class=\"n\">b</span> <span class=\"o\">=</span> <span class=\"n\">args</span><span class=\"p\">[</span><span class=\"mi\">0</span><span class=\"p\">],</span> <span class=\"n\">args</span><span class=\"p\">[</span><span class=\"o\">-</span><span class=\"mi\">1</span><span class=\"p\">]</span>\n",
" <span class=\"k\">if</span> <span class=\"n\">s</span><span class=\"o\">.</span><span class=\"n\">op</span> <span class=\"o\">==</span> <span class=\"s1\">'==>'</span><span class=\"p\">:</span>\n",
" <span class=\"k\">return</span> <span class=\"n\">b</span> <span class=\"o\">|</span> <span class=\"o\">~</span><span class=\"n\">a</span>\n",
" <span class=\"k\">elif</span> <span class=\"n\">s</span><span class=\"o\">.</span><span class=\"n\">op</span> <span class=\"o\">==</span> <span class=\"s1\">'<=='</span><span class=\"p\">:</span>\n",
" <span class=\"k\">return</span> <span class=\"n\">a</span> <span class=\"o\">|</span> <span class=\"o\">~</span><span class=\"n\">b</span>\n",
" <span class=\"k\">elif</span> <span class=\"n\">s</span><span class=\"o\">.</span><span class=\"n\">op</span> <span class=\"o\">==</span> <span class=\"s1\">'<=>'</span><span class=\"p\">:</span>\n",
" <span class=\"k\">return</span> <span class=\"p\">(</span><span class=\"n\">a</span> <span class=\"o\">|</span> <span class=\"o\">~</span><span class=\"n\">b</span><span class=\"p\">)</span> <span class=\"o\">&</span> <span class=\"p\">(</span><span class=\"n\">b</span> <span class=\"o\">|</span> <span class=\"o\">~</span><span class=\"n\">a</span><span class=\"p\">)</span>\n",
" <span class=\"k\">elif</span> <span class=\"n\">s</span><span class=\"o\">.</span><span class=\"n\">op</span> <span class=\"o\">==</span> <span class=\"s1\">'^'</span><span class=\"p\">:</span>\n",
" <span class=\"k\">assert</span> <span class=\"nb\">len</span><span class=\"p\">(</span><span class=\"n\">args</span><span class=\"p\">)</span> <span class=\"o\">==</span> <span class=\"mi\">2</span> <span class=\"c1\"># TODO: relax this restriction</span>\n",
" <span class=\"k\">return</span> <span class=\"p\">(</span><span class=\"n\">a</span> <span class=\"o\">&</span> <span class=\"o\">~</span><span class=\"n\">b</span><span class=\"p\">)</span> <span class=\"o\">|</span> <span class=\"p\">(</span><span class=\"o\">~</span><span class=\"n\">a</span> <span class=\"o\">&</span> <span class=\"n\">b</span><span class=\"p\">)</span>\n",
" <span class=\"k\">else</span><span class=\"p\">:</span>\n",
" <span class=\"k\">assert</span> <span class=\"n\">s</span><span class=\"o\">.</span><span class=\"n\">op</span> <span class=\"ow\">in</span> <span class=\"p\">(</span><span class=\"s1\">'&'</span><span class=\"p\">,</span> <span class=\"s1\">'|'</span><span class=\"p\">,</span> <span class=\"s1\">'~'</span><span class=\"p\">)</span>\n",
" <span class=\"k\">return</span> <span class=\"n\">Expr</span><span class=\"p\">(</span><span class=\"n\">s</span><span class=\"o\">.</span><span class=\"n\">op</span><span class=\"p\">,</span> <span class=\"o\">*</span><span class=\"n\">args</span><span class=\"p\">)</span>\n",
"</pre></div>\n",
"</body>\n",
"</html>\n"
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" </style>\n",
"</head>\n",
"<body>\n",
"<h2></h2>\n",
"\n",
"<div class=\"highlight\"><pre><span></span><span class=\"k\">def</span> <span class=\"nf\">move_not_inwards</span><span class=\"p\">(</span><span class=\"n\">s</span><span class=\"p\">):</span>\n",
" <span class=\"sd\">"""Rewrite sentence s by moving negation sign inward.</span>\n",
"<span class=\"sd\"> >>> move_not_inwards(~(A | B))</span>\n",
"<span class=\"sd\"> (~A & ~B)"""</span>\n",
" <span class=\"n\">s</span> <span class=\"o\">=</span> <span class=\"n\">expr</span><span class=\"p\">(</span><span class=\"n\">s</span><span class=\"p\">)</span>\n",
" <span class=\"k\">if</span> <span class=\"n\">s</span><span class=\"o\">.</span><span class=\"n\">op</span> <span class=\"o\">==</span> <span class=\"s1\">'~'</span><span class=\"p\">:</span>\n",
" <span class=\"k\">def</span> <span class=\"nf\">NOT</span><span class=\"p\">(</span><span class=\"n\">b</span><span class=\"p\">):</span>\n",
" <span class=\"k\">return</span> <span class=\"n\">move_not_inwards</span><span class=\"p\">(</span><span class=\"o\">~</span><span class=\"n\">b</span><span class=\"p\">)</span>\n",
" <span class=\"n\">a</span> <span class=\"o\">=</span> <span class=\"n\">s</span><span class=\"o\">.</span><span class=\"n\">args</span><span class=\"p\">[</span><span class=\"mi\">0</span><span class=\"p\">]</span>\n",
" <span class=\"k\">if</span> <span class=\"n\">a</span><span class=\"o\">.</span><span class=\"n\">op</span> <span class=\"o\">==</span> <span class=\"s1\">'~'</span><span class=\"p\">:</span>\n",
" <span class=\"k\">return</span> <span class=\"n\">move_not_inwards</span><span class=\"p\">(</span><span class=\"n\">a</span><span class=\"o\">.</span><span class=\"n\">args</span><span class=\"p\">[</span><span class=\"mi\">0</span><span class=\"p\">])</span> <span class=\"c1\"># ~~A ==> A</span>\n",
" <span class=\"k\">if</span> <span class=\"n\">a</span><span class=\"o\">.</span><span class=\"n\">op</span> <span class=\"o\">==</span> <span class=\"s1\">'&'</span><span class=\"p\">:</span>\n",
" <span class=\"k\">return</span> <span class=\"n\">associate</span><span class=\"p\">(</span><span class=\"s1\">'|'</span><span class=\"p\">,</span> <span class=\"nb\">list</span><span class=\"p\">(</span><span class=\"nb\">map</span><span class=\"p\">(</span><span class=\"n\">NOT</span><span class=\"p\">,</span> <span class=\"n\">a</span><span class=\"o\">.</span><span class=\"n\">args</span><span class=\"p\">)))</span>\n",
" <span class=\"k\">if</span> <span class=\"n\">a</span><span class=\"o\">.</span><span class=\"n\">op</span> <span class=\"o\">==</span> <span class=\"s1\">'|'</span><span class=\"p\">:</span>\n",
" <span class=\"k\">return</span> <span class=\"n\">associate</span><span class=\"p\">(</span><span class=\"s1\">'&'</span><span class=\"p\">,</span> <span class=\"nb\">list</span><span class=\"p\">(</span><span class=\"nb\">map</span><span class=\"p\">(</span><span class=\"n\">NOT</span><span class=\"p\">,</span> <span class=\"n\">a</span><span class=\"o\">.</span><span class=\"n\">args</span><span class=\"p\">)))</span>\n",
" <span class=\"k\">return</span> <span class=\"n\">s</span>\n",
" <span class=\"k\">elif</span> <span class=\"n\">is_symbol</span><span class=\"p\">(</span><span class=\"n\">s</span><span class=\"o\">.</span><span class=\"n\">op</span><span class=\"p\">)</span> <span class=\"ow\">or</span> <span class=\"ow\">not</span> <span class=\"n\">s</span><span class=\"o\">.</span><span class=\"n\">args</span><span class=\"p\">:</span>\n",
" <span class=\"k\">return</span> <span class=\"n\">s</span>\n",
" <span class=\"k\">else</span><span class=\"p\">:</span>\n",
" <span class=\"k\">return</span> <span class=\"n\">Expr</span><span class=\"p\">(</span><span class=\"n\">s</span><span class=\"o\">.</span><span class=\"n\">op</span><span class=\"p\">,</span> <span class=\"o\">*</span><span class=\"nb\">list</span><span class=\"p\">(</span><span class=\"nb\">map</span><span class=\"p\">(</span><span class=\"n\">move_not_inwards</span><span class=\"p\">,</span> <span class=\"n\">s</span><span class=\"o\">.</span><span class=\"n\">args</span><span class=\"p\">)))</span>\n",
"</pre></div>\n",
"</body>\n",
"</html>\n"
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"\n",
" </style>\n",
"</head>\n",
"<body>\n",
"<h2></h2>\n",
"\n",
"<div class=\"highlight\"><pre><span></span><span class=\"k\">def</span> <span class=\"nf\">distribute_and_over_or</span><span class=\"p\">(</span><span class=\"n\">s</span><span class=\"p\">):</span>\n",
" <span class=\"sd\">"""Given a sentence s consisting of conjunctions and disjunctions</span>\n",
"<span class=\"sd\"> of literals, return an equivalent sentence in CNF.</span>\n",
"<span class=\"sd\"> >>> distribute_and_over_or((A & B) | C)</span>\n",
"<span class=\"sd\"> ((A | C) & (B | C))</span>\n",
"<span class=\"sd\"> """</span>\n",
" <span class=\"n\">s</span> <span class=\"o\">=</span> <span class=\"n\">expr</span><span class=\"p\">(</span><span class=\"n\">s</span><span class=\"p\">)</span>\n",
" <span class=\"k\">if</span> <span class=\"n\">s</span><span class=\"o\">.</span><span class=\"n\">op</span> <span class=\"o\">==</span> <span class=\"s1\">'|'</span><span class=\"p\">:</span>\n",
" <span class=\"n\">s</span> <span class=\"o\">=</span> <span class=\"n\">associate</span><span class=\"p\">(</span><span class=\"s1\">'|'</span><span class=\"p\">,</span> <span class=\"n\">s</span><span class=\"o\">.</span><span class=\"n\">args</span><span class=\"p\">)</span>\n",
" <span class=\"k\">if</span> <span class=\"n\">s</span><span class=\"o\">.</span><span class=\"n\">op</span> <span class=\"o\">!=</span> <span class=\"s1\">'|'</span><span class=\"p\">:</span>\n",
" <span class=\"k\">return</span> <span class=\"n\">distribute_and_over_or</span><span class=\"p\">(</span><span class=\"n\">s</span><span class=\"p\">)</span>\n",
" <span class=\"k\">if</span> <span class=\"nb\">len</span><span class=\"p\">(</span><span class=\"n\">s</span><span class=\"o\">.</span><span class=\"n\">args</span><span class=\"p\">)</span> <span class=\"o\">==</span> <span class=\"mi\">0</span><span class=\"p\">:</span>\n",
" <span class=\"k\">return</span> <span class=\"bp\">False</span>\n",
" <span class=\"k\">if</span> <span class=\"nb\">len</span><span class=\"p\">(</span><span class=\"n\">s</span><span class=\"o\">.</span><span class=\"n\">args</span><span class=\"p\">)</span> <span class=\"o\">==</span> <span class=\"mi\">1</span><span class=\"p\">:</span>\n",
" <span class=\"k\">return</span> <span class=\"n\">distribute_and_over_or</span><span class=\"p\">(</span><span class=\"n\">s</span><span class=\"o\">.</span><span class=\"n\">args</span><span class=\"p\">[</span><span class=\"mi\">0</span><span class=\"p\">])</span>\n",
" <span class=\"n\">conj</span> <span class=\"o\">=</span> <span class=\"n\">first</span><span class=\"p\">(</span><span class=\"n\">arg</span> <span class=\"k\">for</span> <span class=\"n\">arg</span> <span class=\"ow\">in</span> <span class=\"n\">s</span><span class=\"o\">.</span><span class=\"n\">args</span> <span class=\"k\">if</span> <span class=\"n\">arg</span><span class=\"o\">.</span><span class=\"n\">op</span> <span class=\"o\">==</span> <span class=\"s1\">'&'</span><span class=\"p\">)</span>\n",
" <span class=\"k\">if</span> <span class=\"ow\">not</span> <span class=\"n\">conj</span><span class=\"p\">:</span>\n",
" <span class=\"k\">return</span> <span class=\"n\">s</span>\n",
" <span class=\"n\">others</span> <span class=\"o\">=</span> <span class=\"p\">[</span><span class=\"n\">a</span> <span class=\"k\">for</span> <span class=\"n\">a</span> <span class=\"ow\">in</span> <span class=\"n\">s</span><span class=\"o\">.</span><span class=\"n\">args</span> <span class=\"k\">if</span> <span class=\"n\">a</span> <span class=\"ow\">is</span> <span class=\"ow\">not</span> <span class=\"n\">conj</span><span class=\"p\">]</span>\n",
" <span class=\"n\">rest</span> <span class=\"o\">=</span> <span class=\"n\">associate</span><span class=\"p\">(</span><span class=\"s1\">'|'</span><span class=\"p\">,</span> <span class=\"n\">others</span><span class=\"p\">)</span>\n",
" <span class=\"k\">return</span> <span class=\"n\">associate</span><span class=\"p\">(</span><span class=\"s1\">'&'</span><span class=\"p\">,</span> <span class=\"p\">[</span><span class=\"n\">distribute_and_over_or</span><span class=\"p\">(</span><span class=\"n\">c</span> <span class=\"o\">|</span> <span class=\"n\">rest</span><span class=\"p\">)</span>\n",
" <span class=\"k\">for</span> <span class=\"n\">c</span> <span class=\"ow\">in</span> <span class=\"n\">conj</span><span class=\"o\">.</span><span class=\"n\">args</span><span class=\"p\">])</span>\n",
" <span class=\"k\">elif</span> <span class=\"n\">s</span><span class=\"o\">.</span><span class=\"n\">op</span> <span class=\"o\">==</span> <span class=\"s1\">'&'</span><span class=\"p\">:</span>\n",
" <span class=\"k\">return</span> <span class=\"n\">associate</span><span class=\"p\">(</span><span class=\"s1\">'&'</span><span class=\"p\">,</span> <span class=\"nb\">list</span><span class=\"p\">(</span><span class=\"nb\">map</span><span class=\"p\">(</span><span class=\"n\">distribute_and_over_or</span><span class=\"p\">,</span> <span class=\"n\">s</span><span class=\"o\">.</span><span class=\"n\">args</span><span class=\"p\">)))</span>\n",
" <span class=\"k\">else</span><span class=\"p\">:</span>\n",
" <span class=\"k\">return</span> <span class=\"n\">s</span>\n",
"</pre></div>\n",
"</body>\n",
"</html>\n"
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"text/plain": [
"<IPython.core.display.HTML object>"
]
},
"metadata": {},
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],
"source": [
"psource(eliminate_implications)\n",
"psource(move_not_inwards)\n",
"psource(distribute_and_over_or)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Let's convert some sentences to see how it works\n"
]
},
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"execution_count": 32,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"((A | ~B) & (B | ~A))"
]
},
"execution_count": 32,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"A, B, C, D = expr('A, B, C, D')\n",
"to_cnf(A |'<=>'| B)"
]
},
{
"cell_type": "code",
"execution_count": 33,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"((A | ~B | ~C) & (B | ~A) & (C | ~A))"
]
},
"execution_count": 33,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"to_cnf(A |'<=>'| (B & C))"
]
},
{
"cell_type": "code",
"execution_count": 34,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"(A & (C | B) & (D | B))"
]
},
"execution_count": 34,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"to_cnf(A & (B | (C & D)))"
]
},
{
"cell_type": "code",
"execution_count": 35,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"((B | ~A | C | ~D) & (A | ~A | C | ~D) & (B | ~B | C | ~D) & (A | ~B | C | ~D))"
]
},
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"to_cnf((A |'<=>'| ~B) |'==>'| (C | ~D))"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Coming back to our resolution problem, we can see how the `to_cnf` function is utilized here"
]
},
{
"cell_type": "code",
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"metadata": {},
"outputs": [
{
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"\n",
"<div class=\"highlight\"><pre><span></span><span class=\"k\">def</span> <span class=\"nf\">pl_resolution</span><span class=\"p\">(</span><span class=\"n\">KB</span><span class=\"p\">,</span> <span class=\"n\">alpha</span><span class=\"p\">):</span>\n",
" <span class=\"sd\">"""Propositional-logic resolution: say if alpha follows from KB. [Figure 7.12]"""</span>\n",
" <span class=\"n\">clauses</span> <span class=\"o\">=</span> <span class=\"n\">KB</span><span class=\"o\">.</span><span class=\"n\">clauses</span> <span class=\"o\">+</span> <span class=\"n\">conjuncts</span><span class=\"p\">(</span><span class=\"n\">to_cnf</span><span class=\"p\">(</span><span class=\"o\">~</span><span class=\"n\">alpha</span><span class=\"p\">))</span>\n",
" <span class=\"n\">new</span> <span class=\"o\">=</span> <span class=\"nb\">set</span><span class=\"p\">()</span>\n",
" <span class=\"k\">while</span> <span class=\"bp\">True</span><span class=\"p\">:</span>\n",
" <span class=\"n\">n</span> <span class=\"o\">=</span> <span class=\"nb\">len</span><span class=\"p\">(</span><span class=\"n\">clauses</span><span class=\"p\">)</span>\n",
" <span class=\"n\">pairs</span> <span class=\"o\">=</span> <span class=\"p\">[(</span><span class=\"n\">clauses</span><span class=\"p\">[</span><span class=\"n\">i</span><span class=\"p\">],</span> <span class=\"n\">clauses</span><span class=\"p\">[</span><span class=\"n\">j</span><span class=\"p\">])</span>\n",
" <span class=\"k\">for</span> <span class=\"n\">i</span> <span class=\"ow\">in</span> <span class=\"nb\">range</span><span class=\"p\">(</span><span class=\"n\">n</span><span class=\"p\">)</span> <span class=\"k\">for</span> <span class=\"n\">j</span> <span class=\"ow\">in</span> <span class=\"nb\">range</span><span class=\"p\">(</span><span class=\"n\">i</span><span class=\"o\">+</span><span class=\"mi\">1</span><span class=\"p\">,</span> <span class=\"n\">n</span><span class=\"p\">)]</span>\n",
" <span class=\"k\">for</span> <span class=\"p\">(</span><span class=\"n\">ci</span><span class=\"p\">,</span> <span class=\"n\">cj</span><span class=\"p\">)</span> <span class=\"ow\">in</span> <span class=\"n\">pairs</span><span class=\"p\">:</span>\n",
" <span class=\"n\">resolvents</span> <span class=\"o\">=</span> <span class=\"n\">pl_resolve</span><span class=\"p\">(</span><span class=\"n\">ci</span><span class=\"p\">,</span> <span class=\"n\">cj</span><span class=\"p\">)</span>\n",
" <span class=\"k\">if</span> <span class=\"bp\">False</span> <span class=\"ow\">in</span> <span class=\"n\">resolvents</span><span class=\"p\">:</span>\n",
" <span class=\"k\">return</span> <span class=\"bp\">True</span>\n",
" <span class=\"n\">new</span> <span class=\"o\">=</span> <span class=\"n\">new</span><span class=\"o\">.</span><span class=\"n\">union</span><span class=\"p\">(</span><span class=\"nb\">set</span><span class=\"p\">(</span><span class=\"n\">resolvents</span><span class=\"p\">))</span>\n",
" <span class=\"k\">if</span> <span class=\"n\">new</span><span class=\"o\">.</span><span class=\"n\">issubset</span><span class=\"p\">(</span><span class=\"nb\">set</span><span class=\"p\">(</span><span class=\"n\">clauses</span><span class=\"p\">)):</span>\n",
" <span class=\"k\">return</span> <span class=\"bp\">False</span>\n",
" <span class=\"k\">for</span> <span class=\"n\">c</span> <span class=\"ow\">in</span> <span class=\"n\">new</span><span class=\"p\">:</span>\n",
" <span class=\"k\">if</span> <span class=\"n\">c</span> <span class=\"ow\">not</span> <span class=\"ow\">in</span> <span class=\"n\">clauses</span><span class=\"p\">:</span>\n",
" <span class=\"n\">clauses</span><span class=\"o\">.</span><span class=\"n\">append</span><span class=\"p\">(</span><span class=\"n\">c</span><span class=\"p\">)</span>\n",
"</pre></div>\n",
"</body>\n",
"</html>\n"
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"<IPython.core.display.HTML object>"
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],
"source": [
"psource(pl_resolution)"
]
},
{
"cell_type": "code",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"(True, False)"
]
},
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"pl_resolution(wumpus_kb, ~P11), pl_resolution(wumpus_kb, P11)"
]
},
{
"cell_type": "code",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"(False, False)"
]
},
"metadata": {},
"output_type": "execute_result"